Table of Contents

Last modified on July 12th, 2024

A superset is a set that contains all the elements of another set.

If we have two sets, A and B, we say that A is a superset of B if every element of A is also an element of B. This means if B is a subset of A, then A is the superset of B.

If A = {1, 2, 3, 4, 5, 6, 7} and B = {1, 3, 5}

Since A contains all the elements of B, A is a superset of B.

It is written as **A ⊇ B**

The line under the symbol ‘⊃’ means that set A may also be equal to set B. In such cases, they will be identical sets. However, if set A is a proper superset of set B (where at least one element of A is not in set B), we remove the line and write **A ⊃ B.**

Moreover, when a set is not a superset of a given set, we put a slash through the symbol ‘⊃’ and write **A ⊅ B** (read as A is not a superset of B).

- Since an empty set has no elements, every set is considered a superset of an empty set. For any set A, A ⊃ ɸ
- Every set is a superset of itself. For any set A, A ⊃ A
- The number of supersets of a set is infinite.

A proper superset is a set that contains all the elements of another set and at least one additional element.

Set A is a proper superset of set B if every element of B is also an element of A, and A contains at least one element not in B.

A proper superset is denoted by the symbol ‘⊃.’ If A is a superset of B and A ≠ B, it is expressed as **A ⊃ B**, read as ‘proper or strict superset of’ but ‘NOT equal to.’

If A = {set of quadrilaterals} and B = {set of regular quadrilaterals}

Then, A ⊃ B

Since a set of whole numbers, natural numbers, integers, rational numbers, or irrational numbers are all parts of real numbers, the set of real numbers is the superset of these sets.

ℝ ⊃ ℕ

ℝ ⊃ 𝕎

ℝ ⊃ ℤ

ℝ ⊃ ℚ

ℝ ⊃ ℚ’

An improper superset is a set that contains all the elements of another set, making it exactly equal to the given set.

An improper superset can sometimes be a superset that is not strictly larger than the subset it contains.

A set A is an improper superset of set B if every element of B is also an element of A, including the possibility that A and B are equal.

An improper superset is denoted by the symbol ‘⊇.’ If A is an improper superset of B, it is expressed as **A ⊇ B**, read simply as ‘superset of,’ where the condition A = B is considered.

If A = {1, 3, 5} and B = {1, 3, 5}

Then A ⊇ B

Again, if X = {apple, mango, banana} and Y = {mango, banana, apple}

Then X ⊇ Y

Basis | Superset | Subset |
---|---|---|

Definition | A set consisting of all elements of the original set. | A set consisting of fewer or the same elements as the original set. |

Symbol/Notation | Represented by the notation A ⊇ B (A is a superset of B) or A ⊃ B (A is a proper superset of B). | Represented by the notation A ⊆ B (A is a subset of B) or A ⊂ B(A is a proper subset of B). |

Relation to Superset/Subset | The size of the superset is greater than or equal to the size of the subset. | The size of the subset is less than or equal to the size of the superset. |

Relation to Empty Set | The empty set (ɸ) is a superset of every set. | The empty set (ɸ) is a subset of every set. |

Example | Example:{1, 5, 10} is a superset of {1, 5}. | Example:{1, 5} is a subset of {1, 5, 10}. |

**If A = {2, 4, 6, 8, …, 20} and B = {4, 8, 12, 16, 20}, identify the superset and the subset.**

Solution:

Given A = {2, 4, 6, 8, …, 20} and B = {4, 8, 12, 16, 20}

Since all elements of set B are present in set A, set B is a part of set A.

Thus, A is a superset of B, and B is a subset of A.

**Determine the superset when two sets are A = {x | x ∈ ℕ} and B = {x | x is an odd number}.**

Solution:

Given A = {x | x ∈ ℕ} and B = {x | x is an odd number}

A = {1, 2, 3, 4, 5, 6, 7, …} and B = {1, 3, 5, 7, 9, …}

Here, set A contains all elements of set B, which means B is a part of A.

Thus, A is a superset of B.

Last modified on July 12th, 2024